Minimax Bounds for Sparse PCA with Noisy High-Dimensional Data

• Activity Number: 506
• Type: Invited
• Date/Time: Wednesday, August 7, 2013 : 10:30 AM to 12:20 PM
• Sponsor: IMS
• Author(s): Iain M. Johnstone*+
• Companies: Stanford University
• Keywords: Principal components analysis ; eigenvector ; sparsity ; lower bound ; covariance matrix

Abstract

We study the problem of estimating the leading eigenvectors of a high-dimensional population covariance matrix based on independent Gaussian observations. We establish a lower bound on the minimax risk of estimators under the quadratic loss, in the joint limit as dimension and sample size increase to infinity, under various models of sparsity for the population eigenvectors. The lower bound on the risk points to the existence of different regimes of sparsity of the eigenvectors. We also propose a new method for estimating the eigenvectors by a two-stage coordinate selection scheme. Joint work with Aharon Birnbaum, Boaz Nadler and Debashis Paul.